LIBRARY 

OF  THE 

UNIVERSITY  OF  CALIFORNIA. 


GIFT  OF" 


SCIENCE  {  FUNCTION  }  ANALYSIS 


Man  gewinnt  dadurch  schon  sehr  viel,  wenn  man  eine 
Menge  von  Untersuchungen  unter  die  Formel  einer  ein- 
zigen  Aufgabe  bringen  kann.  Denn  dadurch  erleich- 
tert  man  sich  nicht  allein  selbst  sein  eigenes  Geschaft, 
indem  man  es  sich  genatt  bestimmt,  sondern  auch  jedem 
anderen,  der  es  priifen  will,  das  Urtheil,  ob  wir  unserem 
Vorhaben  ein  Geniige  gethan  haben  oder  nicht. — Kant. 


PROBLEM 

SCIENCE  \  FUNCTION  {  ANALYSIS 

FORMULA 

SCIENCE  I      I  ANALYSIS 

SCIENCE  {  X  }  ANALYSIS 

INDIVIDUAL 

SCIENCE  <      GENERAL      >  ANALYSIS 
UNIVERSAL 

BY 
GECKOS  ASHTON  BLACK,  PH.D. 


NEW  YORK 

PRINTED  FOE  THE  AUTHOR 
1905 


Copyright,  1905,  by 
GEORGE  ASHTON  BLACK 


Published  January,  1905 


THE  DeViNNE  PRESS. 


SCIENCE  {FUNCTION?  ANALYSIS 

SCIENCE  {      }  ANALYSIS 
SCIENCE  I  X  }  ANALYSIS 

f  INDIVIDUAL  1 

SCIENCE  <     GENERAL     >  ANALYSIS 
UNIVERSAL 


U\BRA 

OF  THE 

(  UNIVERSITY  ) 

OF 


PROPAEDEUTIC  TO 
SCIENCE  {FUNCTION}  ANALYSIS 

I     Resolution  of  all  cases  of  the  hitherto  unrecognized  gen- 
eral problem,  Differential  Coefficient  —  Function. 


Dif.  coef. 


Function 


imag. 


f  individual 
real     <     general 
\  universal 


where  differential  coefficient  is  either  imaginary  or  real  ac- 
cording as  function  applies  either  to  any  imaginary  variable 
sufficiently  indicated  by  referred  to  the  logical  deter- 

mination of  imaginary,  or  to  any  real  variable  sufficiently 
indicated  by  x  referred  to  the  logical  determination  of  real ; 
and  where  the  first  complete  determination  of  function  by 
itself  constitutes  the  only  variety  necessary  and  sufficient 
whether  to  differentiate  the  identity  of  function  in  my 
Science  {function}  Analysis,  or  to  integrate  the  difference  of 
function  in  Kant's  enumeration  of  all  moments  of  a  single 
cognition. 

II  Determination  of  the  relation  of  Science  {function} 
Analysis  to  Logic. 

On  reflection,  the  primitive  phrase,  a  single  cognition,  is 
found  to  lead  to  the  complete  determination  of  the  formula 


a  single  cognition  of  any  kind  concept      I 

a  single  necessary  and  universal  cognition  judgment  V 

a  single  necessary  {cognition}  universal  syllogism  I 


where  the  derivative  phrases  determine  the  primitive  phrase 
in  relation  to  all  and  only  the  fundamental  distinctions  of 
thought,  that  is  to  say,  the  fundamental  logical  principles 
of  the  agreement  of  thought  with  itself ;  and  where  the  vari- 
able recognized  in  the  primitive  can  be  replaced  by  its  limit 
recognized  according  to  a  different  mode  in  every  single 
derivative.  This  primitive  logic,  as  the  formula  deserves  to 
be  called,  demonstrates  that  the  accepted  logical  determina- 
tion of  thought  is  not  a  logical  division  but  a  logical  trans- 
formation, and  renders  necessary,  at  least  as  a  problem,  the 
invention  or  discovery  of  a  series  of  three  derivative  logics 
which  shall  by  degrees  extend  the  application  of  every  single 
differential  moment  of  the  primitive  logic  as  far  as  possible, 
that  is  to  say,  over  the  whole  domain  whether  of  a  single 
cognition  or  of  thought.  From  the  fact  that  the  primitive 
application  of  the  third  differential  moment  appears  in  the 
deduced  formula  of  Science  {function}  Analysis  as  the  dif- 
ferentiation of  the  identity  of  function  in  an  integrated 
equation,  I  infer  that  the  rest  of  the  third  derivative  logic 
will  appear  in  the  deduction  of  the  corresponding  work  as 
the  integration  of  the  difference  of  function  in  a  series  of 
differential  equations ;  and  from  the  nature  and  position 
of  the  third  derivative  moment  I  conclude  that  this  third 
derivative  logic,  considered  as  if  it  had  all  appeared,  is  the 
only  logic  that  deserves  to  be  called  definitive. 

GEORGE  ASHTON  BLACK 
New  York,  621  W.  113th  Street 
April  28,  1905 


DEDUCTION  OF 
PKOBLEM  AND  FORMULA 

DEFINITION 

Science  =  necessary  I cognition }  universal  =  the  reso- 
lution of  all  cases  of  a  general  problem  =  Analysis. 

Hence 
Science  j  function  \  Analysis 

Hence 

Science  \         \  Analysis 
Science  \    x    \  Analysis 

(  individual  } 

Science  <     general     >  Analysis 
(   universal  ) 

which  satisfies  the  definition  whether  by  resolving  the  prob- 
lem into  the  singular  case  marked  by  the  only  imaginary 
differential  coefficient  \  \  and  the  general  case  marked  by 
any  real  differential  coefficient  \  x  \  where  x  is  individual 
general  universal  through  all  moments  of  the  logical  deter- 
mination 

(  individual  Y 
real  j     general     >  x 
(  universal  ) 

or  by  determining  the  problem  in  relation  to  all  moments 
of  a  single  necessary  f  cognition  \  universal  according  to 
Kant's  classic  enumeration  of  them  in  their  proper  order 

7 


and  connection ;  for  the  only  three  fundamentally  distinct 
moments  of  the  formula  differ  whether  as  intuition  concept 
idea  according  to  the  rule  ideal— or  as  synthesis  analysis 
dialectic  according  to  the  rule  method— of  Science  \  func- 
tion }  Analysis. 

Thus  the  formula  lays  down  the  Science  as  well  as  the 
Analysis  of  my  original  problem  in  the  shape  of  a  series  of 
derivative  problems,  the  demanded  resolution  of  which,  in 
the  order  and  connection  prescribed  by  the  formula,  is  de- 
veloped in  the  sequel  into  an  organon,  yet  to  be  a  complete 
system,  of  Science  \  universal  J  Analysis. 

NOTE.     The  isolated  variety 


individual 

general 

universal 


is  necessary  and  sufficient  to  integrate  or  satisfy  the  isolated 
differential  equation 


(  intuition  synthesis  } 

ideal  <  concept  analysis    ?  method 

(  idea  dialectic    ) 


to  which  Kant's  enumeration  leads  through  the  recognition 
of  the  enumeration  in  two  equivalent  formulas.   Witness  my 


ideal 


intuition  \          \  synthesis 

concept  j    x    \  analysis 

(  individual  ) 

idea  <     general     >     dialectic 

(  universal  ) 
8 


method 


DEDUCTION  OF 
WOKK  DEMANDED  BY  FORMULA 


A 


POSTULATE 

I 

=  a  plane  equilateral  triangle  the  internal  determination 
of  which  is  only  imaginary  =  the  geometrical  construction 
whether  of  the  concept  Science  or  of  the  concept  Analysis 
referred  to  the  vacant  place  of  any  correlative  concept = 
the  solution  of  one  case  of  the  general  problem,  Science 
{  }  Analysis ; 

/  \=  a  plane  equilateral  triangle  the  internal  determina- 
tion of  which  is  real  in  respect  of  only  the  middle  point 
of  the  altitude  =  the  geometrical  construction  whether  of  the 
concept  Science  or  of  the  concept  Analysis  referred  to  the 
concept  individual  =  the  solution  of  one  case  of  the  general 
problem,  Science  {individuals  Analysis. 

Ill 

Resolution  of  all  cases  of  Science  {individuals  Analysis  = 
solution  of  one  case  of  Science  { general }  Analysis. 

IV 

Resolution  of  all  cases  of  Science  {  general }  Analy sis = solu- 
tion of  one  case  of  Science  {  universal }  Analysis. 


HENCE 

the  work  by  corresponding  stages,  as  follows. 

9 


I  Resolution  of  all  cases  of  Science  j  J  Analysis  achieved 
in  and  through  the  integration  of  a  simple  differential 
equation  derived  from  the  solution  of  one  case. 


A 


of  Science 


Analysis 


synthesis 
analysis 


/  \  dialectic 


method 


of 


ideal 
Science 


intuition 
concept 


idea 


A 


synthesis 
analysis 

dialectic 


method 
of       Analysis 


where  the  imaginary  term,  derived  from  the  imaginary  dif- 
ferential coefficient  in  2,  and  posited  in  the  shape  of  a  blank 
space  before  Science  and  likewise  before  Analysis,  should 
not  be  overlooked ;  for  it  is  necessary  and  sufficient  in  that 
place  to  indicate  the  singular  case  of  my  original  problem, 
evidently  the  simplest  case  possible,  which  is  alone  in  ques- 
tion now. 

10 


NOTE.  2  arises  from  distinction  of  successive  stages  in  the 
reproduction  of  1  by  degrees  according  to  a  constant  rule 
referred  to  Science  {  }  Analysis  in  relation  to  which  the 
whole  of  1  was  postulated.  Thus  the  stages  in  question 

/     /       /\    given  Kant's  terminology,  are  found  to  differ 

whether  as  intuition  concept  idea  according  to  the  rule  ideal 
—  or  as  synthesis  analysis  dialectic  according  to  the  rule 
method  —  of  Science  {  }  Analysis. 

3  arises  from  reference,  in  one  to  one  correspondence,  of  the 
successive  moments  of  two  equivalent  formulas  obtained 
from  2  to  the  same  whole  of  possible  Science  J  }  Analysis, 
which  is  thereby  perfectly  differentiated  a  priori ;  followed 
by  the  integration  of  the  resulting  differential  equation. 
The  second  moment  of  the  integral  represents  the  necessary 
identity  in  all  the  variety  presented  by  the  first  moment. 
For  the  first  moment  may  be  conceived  to  undergo  a  gradual 
shortening  of  unit  line  until  all  difference  in  the  variety  line 
angle  surface  vanishes,  and  nothing  is  left  but  the  identity 
in  the  same,  in  the  image  of  a  point.  So  conceived  the  first 
moment  stands  to  the  second  in  the  relation  of  variable  to 
limit.  The  third  moment  of  the  integral  represents  the 
necessary  reference  of  all  the  variety  presented  by  the  first 
moment,  to  the  representation  whether  of  the  variable  or 
limit  in  question. 


11 


II  Resolution  of  all  cases  of  Science  findividual|  Analysis 
achieved  in  and  through  the  integration  of  a  simple  differ- 
ential equation  derived  from  the  solution  of  one  case. 


intuition 


concept 


A  idea 


synthesis 
analysis 


dialectic 


ideal 


of  Science  \  individual  \  Analysis 


method 


ideal 

of  individual 
Science 


intuition 


idea 


synthesis 


concept  D         analysis 

A         B 


AB 

x 
CD       dialectic 

II 
ABC-D 


method 

of  individual 
Analysis 


12 


Ill  Resolution  of  all  cases  of  Science  \  general  \  Analysis 
achieved  in  and  through  the  integration  of  a  complex  dif- 
ferential equation  derived  from  the  solution  of  one  case. 

A 

c 

D 
A         B 

AB 

x 
CD 

ABC-D 


intuition  } 
concept    £  ideal 
idea          ) 

of  Science  [  general  ]  Analysis 
synthesis  ^ 
analysis    >  method 
dialectic    ) 


method  of  general  Analysis 
synthesis  analysis  dialectic 


3 

<D 

intuition      /    \ 

A 

A 

A 

1 

1 

C 

z 

z 

zZ 

1 

d 

concept         D 
A       B 

D 

Y                 V 

J 

D 
X       Y 

D 
xX  yY 

1 

AB 

idea           CD      } 

II 

base 

£  altitude 

II 

multiplicand 
x 
multiplier 

II 

base  =  area  -+-  yz  alt. 

X  XX 

V2  altitude  =  area  -H   base 

ii  ii  ii 

ABC-D 

ii 
area 

product 

ii  ii  ii 
area  =  area2-f-  area 

NOTE.  The  complex  form  of  the  differential  equation  in  3  is 
rendered  necessary  by  the  specialty  of  the  particular  Science 
j  x  }  Analysis  in  question.  Only  through  the  cross  refer- 
ence of  the  two  equivalent  formulas  obtained  from  2  could 
the  same  whole  of  possible  Science  {  general  ]  Analysis  be 
perfectly  differentiated  a  priori. 

13 


IV  a  Derivation  from  the  solution  of  one  case,  of  a  com- 
pound differential  equation  demanding  the  resolution  of  all 
cases,  of  Science  \  universal  }  Analysis. 


A 

c 

D 
A       B 

AB 

x 

CD 

II 
ABC-D 


Z 

D 

X       Y 

base       multiplicand 


zZ 

D 

xX    yY 

base       =  area  — 


y2  altitude    multiplier     yz  altitude  =  area  -H 


product 


II  II 

area       =  area2-1- 


altitude 

x 
base 

II 
area 


single 

definite 

individual 


manifold 
definable 
general 


synthetical       analytical 


restrictive  intuition  } 

definitive  concept    ?  ideal 

universal  idea          ) 

of  Science  \  universal  \  Analysis 

dialectical  synthesis  } 

"  analysis    >  method 

"  dialectic    ) 


3  Here  belong  the  empty  tables  ABC  which  follow. 
They  perfectly  differentiate  the  whole  of  possible  Science 
5  universal  \  Analysis,  and  constitute  a  compound  differen- 
tial equation  demanding  the  resolution  of  all  cases  of  that 
problem. 


14 


"BTp 


'tre     'u.£s 


'yen1®      "erp      'ire 


"Bip 


15 


"Blp 


'U^S 


"Blp 


'U^S 


"Blp         'U13       ' 


jo  aoHi<aj\[ 


16 


Bip        -ire     'iijCs       Terra      "Btp      -U-B 


"Btp      'Tre 


IV  b  Essay  to  refer  recognized  moments  of  the  demanded 
resolution  of  all  cases  of  Science  5 universal}  Analysis  each 
to  its  proper  place  in  the  formula  demanding  the  resolution. 

On  the  supposition  that  pure  reason,  constant  as  the 
faculty  of  Science  Jf unction  j  Analysis  in  all  rationals  of  all 
times,  but  varied  through  all  moments  of  a  maximum  dif- 
ferentiation and  integration  of  that  function  in  different 
rationals  of  different  times,  has  somewhere  in  some  context 
already  cognized  every  step  in  the  solution  of  every  case  of 
Science  {universal}  Analysis,  but  has  not  yet  recognized 
every  step  in  the  solution  of  every  case  in  its  proper  place  in 
the  resolution  of  all  cases ;  I  propose  to  search  out  all  and  only 
the  cognitions  that  are  the  content  of  that  resolution,  and 
arrange  them  each  in  that  proper  place  as  fixed  for  it  a 
priori  by  Tables  ABC.  To  be  sure  the  task  is  not  for 
only  one  rational,  but  for  every  one  interested  in  the 
development  as  much  as  possible  in  himself  of  the  same 
faculty  that  aforetime  made  the  cognitions,  and  is  now  in 
his  person  called  upon  to  recognize  what  it  has  itself  in  other 
persons  already  produced  according  to  a  fixed  and  ascertained 
formula.  My  own  discovery  of  the  required  cognitions  and 
reference  of  them  to  this  or  that  place  in  the  formula  is  sure 
and  complete  as  regards  the  solution  of  only  one  case  of  the 
general  problem  in  question ;  but  all  that  is  wanting  to  the 
perfection  of  the  demanded  resolution  will  undoubtedly  be 
found,  if  able  men,  and  such  as  are  acquainted  with  what  is 
classic  in  the  use  of  pure  reason,  will  endeavor  to  recognize 
the  missing  cognitions  by  the  general  but  sufficient  marks 
that  relegate  them  to  one  or  another  place  in  the  formula 
in  correlation  with  one  or  another  moment  of  the  singular 
solution. 


18 


"Bip 


"Gip       'TIB     * 


jo 


19 


— 1 1 


III 

111 


III  III 


A 


"«ip 


•feu's     "Bip       'ire 


ill 


"Bip 


SISATVXV  qvsaaAmji  do 
20 


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